11 THE SPECTRAL THEORY OF HUBERT 403 PROBLEMS 1. Show that a continuous operator N on a Hilbert space E is normal if and only if llAr-xllHlAr*-*! for all*eE. 2. Let N be a continuous normal operator on a Hilbert space E. (a) For each x e E, consider the open sets W c C such that there exists a continuous mapping /w of W into E satisfying the relation for all A e W. Show that there exists a largest open set &(x) with this property, that all the functions /w are restrictions of a unique mapping / of Q,(x) into E, and that /is analytic in Q(x). (Use (15.5.6) to reduce to the case where the normal operator N is simple. In this case, E being identified with Lc(Sp(f7), JJL) and x with the class of a function g, the set Q,(x) is the interior of the set of all A e C such that the function £i-»(A- 0-V(D belongs to ^£(Sp(AO, /x).) (b) Put <3>(;c) = C — Q.(x). Show that, for each closed subset M of Sp([7), the space E(M) = <pM(t/)(E) is the set of all x E E such that ®(x) <= M. (Again reduce to the case of a simple normal operator.) (c) Show that every continuous operator V e L(E) which commutes with N also commutes with all the operators g(N), where g e ^c(SpW)» an(* *n particular commutes with N* (Fuglede's theorem). (First show that £l(V- x) => H(;c) for all x e E, by considering the function / defined in (a) and the function Ah- ^-/(A). Using (b), deduce that, for each closed subset M of E, the operator V commutes with the projector PE(M) = <PM(W), and conclude that V commutes with g(N) for all con- tinuous functions g on Sp(7V).) (d) Deduce from (c) that, if NI and N* are commuting normal operators, then NtN2 is normal. 3. Let fx be normalized Haar measure on the group U: \z\ = 1 (so that djji(8) — (2-jr)-1 d6) and let E = L£(^)- Then the operator MM(1C) = U is a unitary operator on E. Every closed subspace of E which is stable under N is either of the form <PM(AO(E), where M c U is of measure < 1 ; or else is of the form q • H2(jU.), where |0| = 1 (Section 15.3, Problem 15) (Beurling's theorem). Deduce that there exist closed subspaces F of E which are stable under U but not under £7* = C/"1, and are such that the orthogonal projector PF does not commute with N (compare with (15.15.3)). 4. Without using Riesz theory (11.4), prove that the spectrum of a compact normal operator has only isolated points, except for the point 0 (reduce to the case of a simple normal operator). 5. Show that the spectrum of the self-adjoint operator H considered in (15.1113) is the interval [0, 2] in R. (Observe that \\H\\ <; 2; to prove that #^> 0, that is to say /» b (H • u | u) 2; 0 for all w e E, consider first the functions u(t) = ettx dx and their linear combinations. To show that every number of the form 2/(l + a2) belongs to the spectrum of H, approximate ga by functions «„#«, where un e jSf2 is ^0 and the sequence (un) is increasing and tends to 1 .) p,'H is a measure with base fc-i . /*„